Eur. Phys. J. C (2017) 77:48 DOI 10.1140/epjc/s10052-017-4615-1 Regular Article - Theoretical Physics Inönü–Wigner contraction and D = 2 + 1 supergravity P. K. Concha1,2,a, O. Fierro3,b, E. K. Rodríguez1,2,c 1 Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Av. Padre Hurtado 750, Viña del Mar, Chile 2 Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Casilla 567, Valdivia, Chile 3 Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile Received: 25 November 2016 / Accepted: 5 January 2017 / Published online: 25 January 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract We present a generalization of the standard cannot always be obtained by rescaling the gauge fields and Inönü–Wigner contraction by rescaling not only the gener- considering some limit as in the (anti)commutation relations. ators of a Lie superalgebra but also the arbitrary constants In particular it is well known that, in the presence of the exotic appearing in the components of the invariant tensor. The Lagrangian, the Poincaré limit cannot be applied to a (p, q) procedure presented here allows one to obtain explicitly the AdS CS supergravity [7]. This difficulty can be overcome Chern–Simons supergravity action of a contracted superal- extending the osp (2, p) ⊗ osp (2, q) superalgebra by intro- gebra. In particular we show that the Poincaré limit can be ducing the automorphism generators so (p) and so (q) [9]. performed to a D = 2 + 1 (p, q) AdS Chern–Simons super- In such a case, the IW contraction can be applied and repro- gravity in presence of the exotic form. We also construct a duces the Poincaré limit leading to a new (p, q) Poincaré new three-dimensional (2, 0) Maxwell Chern–Simons super- supergravity which includes additional so (p)⊕so (q) gauge gravity theory as a particular limit of (2, 0) AdS–Lorentz fields. supergravity theory. The generalization for N = p +q grav- Here, we present a generalization of the IW contraction by itinos is also considered. considering not only the rescaling of the generators but also the constants of the non-vanishing components of an invari- ant tensor. The method introduced here ensures the construc- 1 Introduction tion of any CS action based on a contracted (super)algebra. In particular, we show that the Poincaré limit can be applied The three-dimensional (super)gravity theory represents an to a (p, q) AdS supergravity in the presence of the exotic interesting toy model in order to approach higher-dimensional Lagrangian without introducing extra fields as in Ref. [9]. (super)gravity theories, which are not only more difficult but Subsequently, we apply the method to different (p, q) AdS– also leads to tedious calculations. Additionally, the D = 2+1 Lorentz supergravities whose IW contraction leads to diverse model has the remarkable property to be written as a gauge (p, q) Maxwell supergravities. The possibility to turn the IW theory using the Chern–Simons (CS) formalism [1,2]. In par- contraction into an algebraic operation is not new and has ticular, the three-dimensional supersymmetric extension of already been presented in the context of asymptotic symme- General Relativity [3,4] can be obtained as a CS gravity the- tries and higher spin theories in Ref. [26]. Other interesting ory using (A)dS or Poincaré supergroup. A wide class of results using diverse flat limit contractions in supergravity N -extended Supergravities and further extensions have been can be found in Ref. [27]. studied in diverse contexts in, e.g., [5–25]. At the bosonic level, the Maxwell symmetries have lead The derivation of a supergravity action for a given superal- to interesting gravity theories allowing to recover General gebra is not, in general, a trivial task and its construction is not Relativity from Chern–Simons and Born–Infeld (BI) the- always ensured. On the other hand, several (super)algebras ories [28–31]. On the other hand, the AdS–Lorentz and can be obtained as an Inönü–Wigner (IW) contraction of its generalizations allow one to recover the Pure Lovelock a given (super)algebra [38,39]. Nevertheless, the Chern– [32–34] Lagrangian in a matter-free configuration from CS Simons action based on the IW contracted (super)algebra and BI theories [35,36]. At the supersymmetric level, the Maxwell superalgebra provides a pure supergravity action in a e-mail: [email protected] the MacDowell–Mansouri formalism [37]. More recently, a b e-mail: ofi[email protected] three-dimensional CS action based on the minimal Maxwell c e-mail: [email protected] 123 48 Page 2 of 18 Eur. Phys. J. C (2017) 77 :48 superalgebra has been presented in Ref. [22] using the expan- exotic Lagrangian, which has no Poincaré limit [7], is added sion procedure. Here, we show that the same result can be to the AdS CS supergravity. obtained using our alternative approach. Besides, we show The three-dimensional Chern–Simons action is given by ( , ) that the Maxwell limit can also be applied in a p q enlarged supergravity leading to a (p, q) Maxwell supergravity with ( + ) 2 I 2 1 = k Ad A + A3 , (1) an exotic Lagrangian. CS 3 The organization of the present work is as follows: in Sect. 2, we apply our approach to N = 1 and N = p+qAdS where A corresponds to the gauge connection one-form CS supergravities. In particular, we show that the Poincaré and ... denotes the invariant tensor. In the case of the limit can be applied to (p, q) AdS CS supergravity theories in osp (2|1) ⊗ sp (2) superalgebra, the connection one-form is the presence of the exotic Lagrangian. In Sect. 3, we discuss given by the Inönü–Wigner contraction of an expanded supergravity. In particular, we describe the general scheme. In Sect. 4,we 1 ab ˜ 1 a ˜ 1 α ˜ A = ω Jab + e Pa + √ ψ Qα, (2) apply our procedure to a N = 1 expanded CS supergravity. 2 l l In Sect. 5,wepresenttheCSformulationofthe(2, 0) and ( , ) ˜ ˜ ˜ p q Maxwell supergravities and discuss their relations to where Jab, Pa and Qα are the osp (2|1) ⊗ sp (2) generators. (2, 0) and (p, q) AdS–Lorentz supergravities, respectively. The gauge fields ea, ωab and ψ are the dreibein, the spin Section 6 concludes our work with some comments and pos- connection and the gravitino, respectively. Here, the length sible developments. scale l is introduced purposely in order to have dimensionless ˜ ˜ ˜ generators TA ={Jab, Pa, Qα} such that the connection one- A μ form A = Aμ TAdx must also be dimensionless. Since the a = a μ 2 Inönü–Wigner contraction and the invariant tensor dreibein e eμdx is related to the spacetime metric gμν a b through gμν = eμeν ηab, it must have dimensions of length. a/ The standard Inönü–Wigner contraction [38,39]ofaLie Then the “true” gauge field should√ be considered as e l. ψ/ (super)algebra g consists basically in properly rescaling the In the same way, we consider l as the supersymmetry ψ = ψ μ generators by a parameter σ and applying the limit σ →∞ gauge field since the gravitino μdx has dimensions 1/2 corresponding to a contracted (super)algebra. of (length) . ( | )⊗ ( ) Despite having the proper contracted (super)algebra fol- The (anti)-commutation relations for the osp 2 1 sp 2 lowing the IW scheme, the contracted invariant tensor cannot superalgebra are given by be trivially obtained. This is particularly regrettable since the ˜ ˜ ˜ ˜ ˜ ˜ Jab, Jcd = ηbc Jad − ηac Jbd − ηbd Jac + ηad Jbc, (3) invariant tensor is an essential ingredient in the construction of a Chern–Simons action. ˜ ˜ ˜ ˜ Jab, Pc = ηbc Pa − ηac Pb, (4) In this paper, we present a generalization of the standard ˜ ˜ ˜ Inönü–Wigner contraction considering the rescaling not only Pa, Pb = Jab, (5) of the generators but also of the constants appearing in the ˜ ˜ 1 ˜ ˜ ˜ 1 ˜ invariant tensor. The method introduced here allows one to Jab, Qα =− abQ , Pa, Qα =− a Q , 2 α 2 α obtain the non-vanishing components of the invariant tensor (6) of an IW contracted (super)algebra. Thus, the construction ˜ ˜ 1 ab ˜ a ˜ of any CS action based on an IW contracted (super)algebra is Qα, Qβ =− C Jab − 2 C Pa , αβ αβ ensured. In particular, we apply the method to different (p, q) 2 AdS–Lorentz superalgebras whose IW contraction leads to (7) ( , ) diverse p q Maxwell superalgebras. where C denotes the charge conjugation matrix, α repre- Let us first apply the approach to the AdS supergravity in = 1 , sents the Dirac matrices and ab 2 [ a b]. order to derive the Poincaré supergravity. The non-vanishing components of an invariant tensor for the osp (2|1) ⊗ sp (2) superalgebra are given by ( | ) ⊗ ( ) 2.1 Poincaré and osp 2 1 sp 2 supergravity ˜ ˜ Jab Pc = μ1 abc, (8) As in the bosonic level, the IW contraction of the AdS super- ˜ ˜ Jab Jcd = μ0 (ηadηbc − ηacηbd) , (9) algebra leads to the Poincaré one. Besides, the Poincaré CS ˜ ˜ supergravity action can be obtained considering a particular Pa Pb = μ0ηab, (10) limit after an appropriate rescaling of the fields of the super ˜ ˜ AdS CS action. Nevertheless, in the presence of torsion the Qα Qβ = 2 (μ1 − μ0) Cαβ , (11) 123 Eur. Phys. J. C (2017) 77 :48 Page 3 of 18 48 where μ0 and μ1 are arbitrary constants. Then, considering Let us note that the present approach allows one to trivially the invariant tensor (8)–(11) and the connection one-form in obtain the Poincar é limit from the osp (2|1) ⊗ sp (2) CS the general expression for the D = 3 CS action, we have action.